Optimal. Leaf size=28 \[ -\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {457, 79, 65,
213} \begin {gather*} -\frac {1}{4 \sqrt {x^8+1}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {x^8+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 79
Rule 213
Rule 457
Rubi steps
\begin {align*} \int \frac {1+2 x^8}{x \left (1+x^8\right )^{3/2}} \, dx &=\frac {1}{8} \text {Subst}\left (\int \frac {1+2 x}{x (1+x)^{3/2}} \, dx,x,x^8\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^8\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^8}\right )\\ &=-\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 28, normalized size = 1.00 \begin {gather*} -\frac {1}{4 \sqrt {1+x^8}}-\frac {1}{4} \tanh ^{-1}\left (\sqrt {1+x^8}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 27, normalized size = 0.96
method | result | size |
trager | \(-\frac {1}{4 \sqrt {x^{8}+1}}+\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{x^{4}}\right )}{4}\) | \(27\) |
risch | \(-\frac {1}{4 \sqrt {x^{8}+1}}+\frac {\ln \left (\frac {\sqrt {x^{8}+1}-1}{\sqrt {x^{8}}}\right )}{4}\) | \(29\) |
meijerg | \(\frac {-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {x^{8}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{8}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+8 \ln \left (x \right )\right ) \sqrt {\pi }}{2}}{4 \sqrt {\pi }}+\frac {\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {x^{8}+1}}}{2 \sqrt {\pi }}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{4 \, \sqrt {x^{8} + 1}} - \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (20) = 40\).
time = 0.34, size = 52, normalized size = 1.86 \begin {gather*} -\frac {{\left (x^{8} + 1\right )} \log \left (\sqrt {x^{8} + 1} + 1\right ) - {\left (x^{8} + 1\right )} \log \left (\sqrt {x^{8} + 1} - 1\right ) + 2 \, \sqrt {x^{8} + 1}}{8 \, {\left (x^{8} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 10.75, size = 37, normalized size = 1.32 \begin {gather*} \frac {\log {\left (\sqrt {x^{8} + 1} - 1 \right )}}{8} - \frac {\log {\left (\sqrt {x^{8} + 1} + 1 \right )}}{8} - \frac {1}{4 \sqrt {x^{8} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.80, size = 34, normalized size = 1.21 \begin {gather*} -\frac {1}{4 \, \sqrt {x^{8} + 1}} - \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} + 1\right ) + \frac {1}{8} \, \log \left (\sqrt {x^{8} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.85, size = 20, normalized size = 0.71 \begin {gather*} -\frac {\mathrm {atanh}\left (\sqrt {x^8+1}\right )}{4}-\frac {1}{4\,\sqrt {x^8+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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